BLOCKS OF QUOTIENTS OF MACKEY
ALGEBRAS
a thesis submitted to
the graduate school of engineering and science
of bilkent university
in partial fulfillment of the requirements for
the degree of
master of science
in
mathematics
By
Elif Do˘
gan Dar
August, 2015
BLOCKS OF QUOTIENTS OF MACKEY ALGEBRAS By Elif Do˘gan Dar
August, 2015
We certify that I have read this thesis and that in our opinion it is fully adequate, in scope and in quality, as a thesis for the degree of Master of Science.
Assoc. Prof. Dr. Laurence John Barker (Advisor)
Prof. Dr. Mahmut Kuzucuo˘glu
Prof. Dr. Erg¨un Yal¸cın
Approved for the Graduate School of Engineering and Science:
Prof. Dr. Levent Onural Director of the Graduate School
ABSTRACT
BLOCKS OF QUOTIENTS OF MACKEY ALGEBRAS
Elif Do˘gan Dar M.S. in Mathematics
Advisor: Assoc. Prof. Dr. Laurence John Barker August, 2015
We review a theorem by Boltje and K¨ulshammer which states that under certain circumstances the endomorphism ring EndRG(RX) has only one block. We study
the double Burnside ring, the Burnside ring and the transformations between two bases of it, namely the transitive G-set basis and the primitive idempotent basis. We introduce algebras Λ, Λdef and Υ which are quotient algebras of the inflation
Mackey algebra, the deflation Mackey algebra and the ordinary Mackey algebra respectively. We examine the primitive idempotents of Z(Υ). We prove that the algebra Λ has a unique block and give an example where Λdef has two blocks.
Keywords: blocks, double Burnside ring, inflation Mackey algebra, deflation Mackey algebra, ordinary Mackey algebra.
¨
OZET
MACKEY CEB˙IRLER˙IN˙IN B ¨
OL ¨
UM CEB˙IRLER˙IN˙IN
BLOKLARI
Elif Do˘gan Dar Matematik, Y¨uksek Lisans
Tez Danı¸smanı: Assoc. Prof. Dr. Laurence John Barker A˘gustos, 2015
Boltje ve K¨ulshammer’ın bazı ¨ozel ko¸sullar altında ¨ozyapı d¨on¨u¸s¨um halkası EndRG(RX)’in yalnızca bir bloku oldu˘gunu g¨osteren bir teoremini sunaca˘gız.
˙Ikili Burnside halkasını ve Burnside halkasını ¸calı¸saca˘gız ve iki bazı arasındaki d¨on¨u¸s¨um¨u g¨osterece˘giz. Λ, Λdef ve Υ ¸seklinde g¨osterece˘gimiz ¨u¸c cebir
tanımlayaca˘gız. Bu cebirler ¸si¸sirme Mackey cebiri, s¨ond¨urme Mackey cebiri ve adi Mackey cebirinin b¨ol¨um cebirleridir. Ardından Z(Υ)’un ilkel idempotentlerini inceleyece˘giz. Λ cebirinin sadece bir bloku oldu˘gunu g¨osterdikten sonra, Λdef’in
iki blokunun oldu˘gu bir ¨ornek verece˘giz.
Anahtar s¨ozc¨ukler : blok, ikili Burnside halkası, ¸si¸sirme Mackey cebiri, s¨ond¨urme Mackey cebiri, adi Mackey cebiri .
Acknowledgement
First of all, I would like to thank my thesis supervisor, Assoc. Prof. Dr. Laurence J. Barker, for his expertise and knowledge that he generously shared, his valuable guidance, patience and encouragement. Also he was really kind to share his unpublished results. My work would not have been possible without his support. I would like to sincerely thank my father Ender Do˘gan and my mother Nuran Do˘gan for every kind of support they provided, they are the most beautiful people I have ever met.
Also I want to thank my dearest friends Bengi Ruken Yavuz, Merve Demirel and Zeliha Ural for providing information and sharing experiences. I know that without them my life would be surely dull.
Finally, I want to thank my dear husband Salman Ul Hassan Dar for his support and encouragement. His existence is the best gift I have ever received.
Contents
1 Introduction 1
2 Blocks of Endomorphism Rings 3
2.1 Blocks of Endomorphism Rings . . . 3
3 Double Burnside Ring 6 3.1 Bisets and the Double Burnside Ring . . . 6
3.2 Two Bases of the Burnside Algebra . . . 9
4 The Blocks of Λ and Λdef 12 4.1 Fundementals . . . 12
4.2 The Blocks of Υ . . . 14
4.3 The Blocks of Λ . . . 20
Chapter 1
Introduction
In this thesis we focus on finding the blocks of some specific algebras. Throughout the thesis, K will denote a field with characteristic zero and K a finite set of finite groups that is closed under subquotients up to isomorphism. The Burnside ring B(G) of a finite group G, introduced by Dress [1], is the Grothendieck group of the category of finite G-sets with multiplication coming from direct product. In Chapter 2, we give definition of a block and a theorem of Boltje and K¨ulshammer [2] which states that under certain circumstances EndRG(RX) has
only one block.
In Chapter 3, we give the definition of bisets and the double Burnside ring. Also we see that the Burnside ring has two bases, namely the transitive G-set basis and the primitive idempotent basis. In addition, we will give the transformation between these two bases which is found by Gluck [5] and Yoshida [6], indepen-dently.
We have the inflation Mackey category, BB = BBK, generated by ordinary induc-tions, ordinary restrictions and inflations. We also consider the deflation Mackey category , BC = BC
K, similarly defined.
Let⊕KB = M
F,G∈K
of classifying the blocks of ⊕KBB is equivalent to the problem of classifying the
blocks of ⊕KBC, because these two are opposite algebras of each other up to
isomorphism and therefore they have isomorphic centres. However this problem is hard. In this thesis we will consider the quotient algebras Λ and Λdef,
de-fined below, instead of those two algebras. Now let KB = M
G∈K
KB(G), which is a module of the quiver algebra ⊕KB. Let ρ: ⊕KB → EndK( KB) be the representation associated with this module . We introduce the algebras
Λ = ΛK:= ρ(⊕KBB)
Λdef = ΛdefK := ρ(⊕KBC).
In Chapter 4, we show that Λ has a unique block. Also, we will give an example where Λdef has two blocks.
Chapter 2
Blocks of Endomorphism Rings
2.1
Blocks of Endomorphism Rings
Let R be a unital ring. Recall that an idempotent of R is an element e such that e2 = e. Also we call an idempotent e primitive if it cannot be written as a sum of
two orthogonal idempotents. In other words, e cannot be written as e = e1+ e2
such that e1e2 = 0 = e2e1 where e1 and e2 are nonzero idempotents. We define a
block of R to be a primitive idempotent of Z(R).
Remark 2.1. Let A be a finite dimensional algebra over a field. Then 1 =
n
P
i=1
ei
where the ei’s are the primitive idempotents of Z(A). Then
A =
n
M
i=1
Aei
as a direct sum of algebras.
Remark 2.2. Let Λ be a ring and let 1Λ = e1+ e2+ · · · + en be a decomposition
of 1Λ into primitive idempotents e1, . . . , en of Λ. Then every central idempotent
e of Λ is equal to the subsum e = P
i∈I
ei where I denotes the set of all elements
i ∈ {1, 2, . . . n} satisfying eie = ei.
In fact, for an arbitrary i , eei and (1Λ− e)ei are orthogonal idempotents whose
sides of the equation 1Λ= e1+ e2+ · · · + enwith e, we get the desired expression
for e.
The next lemma and theorem is from Boltje and K¨ulshammer [2].
Lemma 2.3 (Boltje-K¨ulshammer). Let G be a finite group, R be an integral domain and X be a transitive G-set. If no prime divisor of |X| is invertible in R, then the RG- permutation module RX is indecomposable.
Proof. We may assume that |X| 6= 1. Assume that RX = ML N is a direct sum decomposition into RG-submodules and assume that M 6= {0}. Then M and N are finitely generated free R-modules and they have a well defined R-rank. Let x ∈ X and H := stabG(x) and let p be a prime divisor of |X| = [G : H].
Since pR 6= R, there exists a maximal ideal P of R such that p ∈ P . Then F := R/P is a field of characteristic p. Let F denote an algebraic closure of F . Then F X ∼= (F ⊗RM ) ⊕ (F ⊗RN ) where F X, F ⊗RM and F ⊗RN are
relatively H-projective F G-modules. By Green’s Indecomposibility Theorem [3], the p-part [G : H]p = |X|p of |X| divides
dimF(F ⊗RM ) = dimF(F ⊗RM ) = rkRp(Rp⊗RM ) = rkR(M )
Since p is arbitrary, we conclude that |X| divides rkR(M ). But,
0 6= rkR(M ) ≤ rkR(M ) + rkR(N ) = rkR(M ⊕ N ) = rkR(RX) = |X|,
which implies rkR(M ) = |X| and rkR(N ) = 0. Thus, N = 0 and M = RX.
Theorem 2.4 (Boltje-K¨ulshammer). Let G be a finite group, X be a finite G-set and R be an integral domain. Assume that, for every x ∈ X and for every prime divisor p of [G : stabG(x)], one has {0} 6= pR 6= R. Then the ring EndRG(RX)
has a unique block.
Proof. Let K denote the field of fractions of R. We decompose X into G-orbits, X = X1F · · · F Xn, and obtain decompositions
into RG-submodules and KG-submodules, respectively. We decompose KXi, for
each i = 1, 2, . . . , n, into indecomposable KG- submodules: KXi = Vi1⊕ · · · ⊕ V
ri
i . (1.2)
We may assume that V1
i ∼= K, the trivial KG-module. In fact, the hypothesis
on R and X implies that |Xi| 6= 0 in K. This implies that ι : K −→ KXi,
1 7−→ |Xi|−1 P x∈Xi
x and π : KXi −→ K, x 7−→ 1 are KG-module
homomor-phisms with π ◦ ι = idK, so that K is isomorphic to a direct summand of KXi.
Let ei ∈ EndRG(RX) denote the idempotent which is the projection onto the i-th
component in the first decomposition in 1.1. Then ei is primitive in EndRG(RX)
by Lemma 2.3. We view EndRG(RX) as a subring of EndKG(KX) via the
canon-ical embedding and decompose ei in EndKG(KX) further into primitive
idempo-tents corresponding to the decomposition in 1.2. ei = e
(1)
i + · · · + e (ri)
i .
Altogether we have a primitive decomposition 1 = (e(1)1 + e(2)1 + · · · + e(r1) 1 ) + · · · + (e (1) n + e (2) n + · · · + e (rn) n ) (1.3)
in EndKG(KX). Now let e be a non-zero central idempotent of EndRG(RX).
Since 1 = e1 + · · · + en is a primitive decomposition of 1 in EndRG(RX), we
have e = P
i∈I
ei for some ∅ 6= I ⊆ {1, 2, . . . n} by Remark 2.2. Since e is also a
central idempotent of EndKG(KX), it is also a subsum of the decomposition in
(1.3). Since ∅ 6= I ⊆ {1, 2, . . . n}, there exists an element i ∈ I, and we have eie = ei. This implies that e
(1) i e = e
(1)
i . For every j ∈ {1, 2, . . . n} there exists
an isomorphism α : KX −→ KX such that αe(1)i α−1 = e(1)j . The equation e(1)i e = e(1)i implies
e(1)j = αe(1)i α−1 = αe(1)i eα−1 = αe(1)i α−1e = e(1)j e.
This implies that eje 6= 0 and Remark 2.2 implies that j ∈ I. Thus I =
Chapter 3
Double Burnside Ring
In this chapter we give the theory of bisets which was initiated by Bouc [4] and we define the double Burnside ring. Also we exhibit two bases of the Burnside algebra and give the transformation between them, which was found by Gluck [5] and Yoshida [6] independently.
3.1
Bisets and the Double Burnside Ring
In this section, we explain how general notions of induction and restriction can be expressed using bisets.
Definition 3.1. Let G and H be groups. An (G, H)-biset U is a set with a left G-action and a right H-action such that these actions commute, i.e.,
∀g ∈ G, ∀u ∈ U, ∀h ∈ H, (g.u).h = g.(u.h).
Definition 3.2. Let G and H be finite groups. The double Burnside ring B(G, H) consists of the formal differences of isomorphism classes of finite (G, H)-bisets. The addition is defined to be disjoint union of (G, H)-bisets, and multi-plication is as follows
of (G, H)-biset U and (H, K)-biset V is defined as the set of H orbits of the carte-sian product U × V where the action of H is defined by (u, v).h := (u.h, h−1.v). It is denoted by U ×H V and the H-orbit of (u, v) is denoted by (u,Hv). The set
U ×H V is a (G, K)-biset with the actions g.(u,Hv).k = (g.u,Hv.k).
Definition 3.4. Let U be a (G, H)-biset. Then for u ∈ U we define the orbit of u as the set of elements whose form is guh where g ∈ G and h ∈ H.
So we can write U as a disjoint union of its orbits: U = G
u∈G\U/H
GuH
where u runs through the representatives of (G, H) orbits.
Definition 3.5. Let U be (G, H)-biset. U is called transitive if it has only one orbit.
We can see every (G, H)-biset as a (G × H)-set by defining the action as (g, h).u := guh−1.
When (G, H)-biset U has only one orbit, i.e., U is transitive, it is isomorphic to [(G × H)/Lu] where Lu is the stabilizer of any element u of U in G × H, i.e.,
Lu = (G, H)u = {(g, h) ∈ G × H | gu = uh, u ∈ U } .
The isomorphism is
(g, h)Lu ∈ [(G × H)/Lu] → guh−1 ∈ U.
Since every (G, H)-biset is a disjoint union of transitive (G, H)-bisets, the double Burnside ring is the free Z module whose generators are the set of isomorphism classes of transitive (G, H)-bisets, ie,
B(G, H) = M L≤G,HG×H Z G × H L .
Given a group homomorphism α : H ← G, we define transitive morphisms in-duction as an (H, G)-biset such that,
HindαG = [H × G/{(α(g), g) : g ∈ G}]
and restriction as a (G, H)-biset such that
GresαH := [G × H/{(g, α(g)) : g ∈ G}] .
When α is injective, following [7], we callHindαG an ordinary induction andGresαH
an ordinary restriction. When α is surjective we write HdefαG =HindαG which we
call deflation and we writeGinfαH =GresαH which we call inflation. When α is an
isomorphism we write HisoαG = HindαG = Gresα
−1
H and call it isogation. When α
is an inclusion we omit the symbol α from the notation, just writing HindG and GresH.
Following the notation of Bouc [8], let
k1(L) := {h ∈ H|(h, 1) ∈ L}
k2(L) := {g ∈ G|(1, g) ∈ L}
p1(L) := {h ∈ H|∃g ∈ G, (h, g) ∈ L}
p2(L) := {g ∈ G|∃h ∈ H, (h, g) ∈ L} .
Definition 3.6. (Star Product) The star product ∗ of two subgroups L ≤ G × H and M ≤ H × K is defined as
L ∗ M = {(g, k) : (g, h) ∈ L and (h, k) ∈ M for some h ∈ H} .
Due to Bouc [8], we have a formula for the product of two bisets.
Theorem 3.7 (Mackey Product Formula, [8]). Let G, H, K be finite groups and let L ≤ G × H and M ≤ H × K. Then
G × H L ×H H × K M = X h∈[p2(L)\H/p1(M )] G × K L ∗(h,1)M .
Also again by Bouc [8], we know that every transitive (G, H)-biset can be written as the composition of the five elementary bisets defined above.
Theorem 3.8 ([8]). Let H and G be groups and L ≤ H × G. H × G
L
=H indDinfD/Ciso ϕ
B/AdefBresG
where D = p1(L), C = k1(L), B = p2(L), A = k2(L) and ϕ : B/A → D/C is an
isomorphism.
3.2
Two Bases of the Burnside Algebra
In this section, we will describe two bases of the Burnside algebra and the trans-formation between these two bases found by Gluck [5] and independently by Yoshida [6].
A finite G-set X is a finite set on which G acts associatively. A G-set X is transitive when there is only one G-orbit in X. In that case, let x ∈ X and let H be the stabilizer of x. Then there is an isomorphism between [X] and [G/H] (the left cosets of H in G). The isomorphism is
gx ∈ X → gH ∈ G/H for g ∈ G.
Let H and K be subgroups of G. Call H and K as G-conjugate, denoted by H =G K, if gHg−1 = K for some g ∈ G. Also, if gHg−1 ⊆ K for some g ∈ G,
we write H ≤G K, and say that H is subconjugate to K.
Given arbitrary G-sets X and Y , we form their disjoint union X qY and cartesian product X × Y , both of which are G-sets. The action of G on X × Y is defined by
g.(x, y) = (gx, gy) for g ∈ G, x ∈ X and y ∈ Y.
Definition 3.9. The Burnside ring of a finite group G, denoted by B(G), is the abelian group generated by the isomorphism classes [X] of finite G-sets X with
addition [X] + [Y ] = [X ] Y ], the disjoint union of the G-sets X and Y . We define the multiplication for G-sets X and Y by [X][Y ] = [X × Y ], the direct product, which makes B(G) a unital commutative ring.
Every G-set can be written as a disjoint union of transitive G-sets. Therefore, {[G/H] : H ≤GG} is a basis for B(G), ie, B(G) =
M
H≤GG
Z [G/H].
Note that, as Z-modules, we can identify B(G, H) = B(G × H), but the product in the previous section is different from the ring multiplication defined in this section.
We define the Burnside algebra over C to be CB(G) = C ⊗ZB(G) =
M
H≤GG
C [G/H] .
LetG
I : I ≤GG be the set of algebra maps CB(G) → C where GI [X] = |XI|
and |XI| is the number of elements fixed by I for a G-set X . The set of primitive idempotents of CB(G) can be written as eGI : I ≤G G where GI(eGI0) = δ(I,I0).
Here δ(I,I0) is 1 if I =G I0 and 0 otherwise. The next well known theorem can be
found in, for instance, Ay¸se Yaman’s thesis [10].
Theorem 3.10. eGI : I ≤G G gives another basis for CB(G).
The table of marks, which we now define, is the transformation matrix from co-ordinates with respect to the basis {[G/U ] : U ≤G G} to coordinates with respect
to the basis eG
I : I ≤G G . Detailed information about it can be found in Ay¸se
Yaman’s thesis [10].
Definition 3.11 (the table of marks). The matrix MG = (mG(I, U ))I,U ≤GG with
rows and columns indexed by representatives of the conjugacy classes of the subgroups of G, is called the Table of Marks where
mG(I, U ) = GI [G/U ] = | {gU ⊆ G : IgU = gU } | = | {g ∈ G : I ≤
g U } |/|U |.
We will use the transformation between these two bases in the next chapter, the transformations are [G/U ] = X I≤GG mG(I, U )eGI and e G I = X U ≤GG m−1G (U, I) [G/U ]. Remark 3.12. Let x be in CB(G), then x can be written as,
x = X
I≤GG
Chapter 4
The Blocks of Λ and Λ
def
In this chapter we will introduce algebras Λ, Λdef and Υ. We will show that Λ has a unique block after classifying the blocks of Υ. Also we will give an example where Λdef has two blocks. Our account is influenced by Barker and draws some
parts from his unpublished notes.
4.1
Fundementals
In the previous chapter we defined the double Burnside ring. In this section we introduce algebras Λ, Λdef and Υ. Let K be a field with characteristic zero
and K be a finite set of finite groups that is closed under subquotients up to isomorphism, i.e., if K E H ≤ G ∈ K then an isomorphic copy of H/K belongs to K.
Definition 4.1. BK is the full subcategory of the biset category such that
Obj(BK) = K and the Z-module of morphisms F ← G in BK is B(F, G) =
B(F × G) where the composition operation B(F, G) × B(G, H) → B(F, H) given by taking G-orbits of direct products. This category is generated by ordinary re-strictions, ordinary inductions, deflations, inflations and isogations by Theorem 3.8.
Definition 4.2. (the inflation Mackey category) BB = BB
K is the subcategory of
BK such that the morphisms are generated by inflations, ordinary inductions and
ordinary restrictions. The category BB is called the inflation Mackey category
for K. Here, the transitive morphisms [(F × G)/I] are such that k2(I) = 1. By
Theorems 3.7 and 3.8, we have, for some epimorphism τI : p1(I) → p2(I),
[(F × G)/I] =Findp1(I)infτpI2(I)resG.
These transitive morphisms comprise a basis for BB(F, G).
Definition 4.3. (the deflation Mackey category) BC = BC
K is the subcategory of
BK such that the morphisms are generated by deflations,ordinary inductions and
ordinary restrictions. The category BCis called the deflation Mackey category for K. Here, the transitive morphisms [(F × G)/I] are such that k1(I) = 1. We
have, for some epimorphism τI : p1(I) ← p2(I),
[(F × G)/I] =Findp1(I)defτpI2(I)resG.
These transitive morphisms comprise a basis for BC(F, G).
Definition 4.4. (the ordinary Mackey category) B∆ = BK∆ is the subcategory of BK such that the morphisms are generated by ordinary inductions and ordinary
restrictions. The category B∆ is called the ordinary Mackey category for K.
Here, transitive morphisms [(F × G)/I] are such that k1(I) = 1 = k2(I). We
have, for some isomorphism τI : p1(I) → p2(I),
[(F × G)/I] =Findp1(I)isoτpI2(I)resG.
These transitive morphisms comprise a basis for B∆(F, G).
Now we consider the problem of classifying the blocks of the category KBB for
given K and K, we mean, the blocks of the algebra
⊕
KBB = M
F,G∈K
KBB(F, G).
This is equivalent to the problem of classifying the blocks of ⊕KBC because
these two are opposite algebras of each other up to isomorphism and there-fore they have isomorphic centres. However this problem is hard. In this the-sis we will consider quotients of these instead of these two algebras. Now let
KB = KBK =
M
G∈K
KB(G). We make KB a ⊕KB-module, via the evident isomorphism KB ∼=M
G∈K
KB(G, 1).
Let ρ: ⊕KB → EndK( KB) be the representation associated with this module. We introduce the algebras
Λ = ΛK:= ρ(⊕KBB)
Λdef = ΛdefK := ρ(⊕KBC) and Υ = ΥK := ρ(⊕KB∆).
4.2
The Blocks of Υ
In this section we will investigate the blocks of Υ which will be used in the next section to find the blocks of Λ.
For a finite-dimensional algebra A, the Jacobson radical J (A) is the unique maxi-mal nilpotent ideal of A, i.e., it is the unique maximaxi-mal ideal such that there exists a natural number k satisfying (J (A))k = 0. Also, it is well known that J (A) is
the unique minimal ideal such that A/J (A) is semisimple. Theorem 4.5. We have ⊕KBB = ⊕
KB∆⊕ J(⊕KBB) and ⊕KBC = ⊕KB∆⊕ J (⊕KBC). In particular ⊕KB∆ is semisimple.
Proof. Proof of this theorem can be found in the paper of Barker [9, Theorem 5.3].
Now we will give an alternative proof to a lemma which can be found in a paper of Boltje-K¨ulshammer [2, Theorem 5.2].
Lemma 4.6. Let A be a unital ring and suppose that A = B ⊕ N where B is a unital subring with 1A = 1B and N is a nilpotent ideal. Then every idempotent
Proof. Let a be an idempotent of Z(A). Since a is an idempotent we have a2 = a
and therefore a = ai for every positive integer i. We write a = b + n for some
b ∈ B and n ∈ N . Again since a is an idempotent we have b + n = (b + n)2 = b2+ nb + bn + n2
b2 ∈ B since B is a subring and nb + bn + n2 ∈ N since N is an ideal. Since we
have a direct sum, these give b2 = b and therefore bi = b for every positive integer i.
Since N is nilpotent nk = (a − b)k = 0 for some positive integer k. This with
ai = a for i = 1, 2, . . . , k gives 0 = (a − b)k= ak+ k k − 1 ak−1(−b) + k k − 2 ak−2(−b)2+ · · · + (−b)k = a + k k − 1 a(−b) + k k − 2 a(−b)2+ · · · + (−b)k. If we multiply both sides with a and reduce the powers of a again, we get
0 = a(a − b)k= ak+1+ k k − 1 ak(−b) + k k − 1 ak−2(−b)2+ · · · + a(−b)k = a + k k − 1 a(−b) + k k − 2 a(−b)2+ · · · + a(−b)k. This gives us a(−b)k= (−b)k which implies with the fact that bi = b , ab = b.
We have ab = b which gives (b + n)b = b2+ nb = b. Since b2 = b, we get nb = 0.
Now since b + n = a is in Z(A), we have
bn + n2 = (b + n)n = an = na = n(b + n) = nb + n2 which implies bn = nb.
Since a = b + n is an idempotent, we have (b + n)2 = b + n. Since bn = nb = 0 we
have b2+ n2 = b + n which gives n2 = n. Therefore ni = n for all positive integer
i. Therefore we have n = nk = 0 which means a = b + n = b.
Definition 4.7. Let G and H be groups. If U is an (H, G)-biset, then the opposite biset Uop is the (G, H)-biset equal to U as a set, with actions defined by
Definition 4.8. If G and H are groups, and L is a subgroup of H × G, then the opposite subgroup L◦ is the subgroup of G × H defined by
L◦ = {(g, h) ∈ G × H | (h, g) ∈ L} . Corollary 4.9. Every idempotent of Z(⊕KBB) and Z(⊕
KBC) belongs to Z(⊕KB∆).
Proof. By using Theorem 4.5, and Lemma 4.6 we get the result for ⊕KBB. Now
take an idempotent e = X L≤(G×G) λL G × G L
from Z(⊕KBC) where G ∈ K and
λL ∈ K. Define φ : ⊕KBC → ⊕KBB , x → x◦ to be the linear map such that
G × G L → G × G L◦
. Then φ(e)φ(e) = φ(e2) = φ(e).
Also if λL 6= 0 then k1(L) = 1, and k2(L◦) = 1 which implies φ(e) is an idempotent
which belongs to Z(⊕KBB). Then by the first part φ(e) belongs to Z(⊕
KB∆). Therefore, e = φ(φ(e)) also belongs to Z(⊕KB∆) because opposite of the ordinary
Mackey category is itself.
Corollary 4.10. The algebra Υ is semisimple and every idempotent of Z(Λ) and Z(Λdef) belongs to Z(Υ).
Proof. By Theorem 4.5, we have ρ(⊕KBB) = ρ(⊕
KB∆) + ρ(J (⊕KBB)). ρ(J (⊕KBB)) is a nilpotent ideal by being the image of a nilpotent ideal
un-der ρ. Since Υ is semisimple, the intersection of Υ with any nilpotent ideal must be zero. Therefore Υ ∩ ρ(J (⊕KBB)) = 0. The result now follows from Lemma 4.6.
Lemma 4.11, 4.14 and 4.17 can be found in the paper by Yoshida [6]. Lemma 4.11. Given finite groups H ≤ G ≥ I , then
HresG(eGI) =
X
I0≤
HH:I0=GI
Proof. Since HresG(eGI) ∈ CB(H) by Remark 3.12 we have, HresG(eGI) = X I0≤ HH HI0(HresG(eGI))eHI0.
Note that for J ≤ H ≤ G and for any G-set X, we have HJ(HresG[X]) = GJ [X] .
By using these, we get,
HresG(eGI) = X I0≤ HH HI0(HresG(eGI))eHI0 = X I0≤ HH GI0(eGI))eHI0 = X I0≤ HH:I0=GI eHI0.
Lemma 4.12 (Mackey Formula, [8]). Let H and K be subgroups of G. Then
KresGindH =
X
g∈[K\G/H]
KindK∩gHcong
Kg∩HresH
where [K \ G/H] is a set of representatives of (K, H)-double cosets in G and cong
is the group isomorphism induced by conjugation by g.
Proof. By Theorem 3.7, we have
KresGindH = K × G {(k, k) : k ∈ K} G × H {(h, h) : h ∈ H} = X g∈[K\G/H] K × H {(k, k) : k ∈ K} ∗(g,1){(h, h) : h ∈ H} = X g∈[K\G/H] K × H {(gh, h) : h ∈ Kg∩ H} = X g∈[K\G/H] K × (K ∩gH) {(gh,gh) :gh ∈ K ∩gH} (K ∩gH) × (Kg∩ H) {(gh, h) : h ∈ Kg∩ H} (Kg ∩ H) × H {(h, h) : h ∈ Kg∩ H} = X g∈[K\G/H] KindK∩gHcong Kg∩HresH. Lemma 4.13. For H ≤ G, GindH(eHH) = |NG(H) : H|eGH.
Proof. Take any K ≤ G, we have GK(GindH(eHH)) = K K(KresGindH(eHH)). By using Lemma 4.12, = KK( X KgH⊆G
KindK∩gHcongKg∩HresH(eHH)).
By the Lemma 4.11 we know that restriction of eH
H to any proper subset of H is
zero. Therefore Kg ∩ H = H and K = K ∩gH, so Kg = H and K = gH.
= X KgH⊆G:K=gH KK(gHcong H(e H H)). Here we have, K K(gHcongH(eHH)) = gH gH(e gH gH) = 1. Therefore, GK(GindH(eHH)) = |NG(H) : H|, if K =G H 0, otherwise Therefore we have; GindH(eHH) = X K≤GG GK(GindH(eHH))eGK = X K≤GG:K=GH |NG(H) : H|eGK = |NG(H) : H|eGH.
Lemma 4.14. Given finite groups J ≤ H ≤ G, then
GindH(eHJ) = |NG(J ) : NH(J )|eGJ.
Proof. Let J ≤ H, we have
|NG(J ) : J |eGJ =GindJ(eJJ) = GindHindJ(eJJ) = |NH(J ) : J |GindH(eHJ)
which implies;
For K ∈ K, let SK be the subspace of the KB spanned by the elements eFI such
that K ∼= I ≤ F ∈ K. As a direct sum of Υ-modules,
KB = M
K∈FK
SK
where K runs over representatives of the isomorphism classes of groups in K. Proposition 4.15. For each K ∈ K, the Υ-submodule SK of KB is simple.
Proof. Let S be a non zero Υ-submodule of SK. Take a non zero element s ∈ S.
We have s = X
G,J
aGJeGJ where G runs over the groups in K and J runs over the G-conjugacy classes of subgroups of G such that J ∼= K. There exists at least one pair (G, J ) such that aG
J 6= 0. Let φ : G ← K be a group monomorphism
such that J = φ(K). By Lemma 4.11, we have
KresφG(e G Js)/(a G J) = e K K ∈ S.
Also let I and F be such that K ∼= I ≤ F ∈ K. Let ψ : F ← G be a group monomorphism such that I = ψ(K). By Lemma 4.14, we have
FindψK(e K
K)/|NF(I)| = eFI ∈ S.
For every K ∈ K, let dK be the K-endomorphism of KB projecting to SK and
annihilating all the other simple modules, i.e., dK(eGI) = bK ∼= IceGI.
Proposition 4.16. The set {dK : K ∈F K} is the set of primitive idempotents
of Z(Υ).
Proof. Since Υ acts faithfully on KB = M
K∈FK
SK, we have
Υ = EndK(S1) ⊕ · · · ⊕ EndK(SK).
Also, since Υ is semisimple, we have Υ =
k
M
i=1
Without loss of generality, one can say Matni(K) ∼= EndK(Si). Now Z(Υ) = k
L
i=1
KIi where Ii is the identity matrix for Matni(K). So primitive idempotents
are (· · · , 0, Ii, 0, · · · ) for i = 1, · · · , k which corresponds to dk’s.
4.3
The Blocks of Λ
We will show that Λ has a unique block by using the blocks of Υ which were found in the previous section.
Lemma 4.17. Given finite groups N E G and N ≤ H ≤ G, then
GinfG/N(e G/N H/N) = X I≤GG:IN =GH eGI.
Proof. Regarding a G/N -set X as a G-set by inflation, we have XI = XIN/N, ie,
GI(X) = G/NIN/N[X]. So we have, GinfG/N(e G/N H/N) = X I≤GG GI(GinfG/N(e G/N H/N))e G I = X I≤GG G/NIN/N(eG/NH/N)eGI = X I≤GG:IN =GH eGI.
Theorem 4.18. The algebra Λ has a unique block.
Proof. By Corollary 4.10 and Proposition 4.16, every idempotent of Z(Λ) is a sum of idempotents having the form dK where K ∈ K. Given an idempotent d
of Z(Λ) and K ∈ K, we have ddK = dK or ddK = 0. We define an equivalence
relation ≡ such that, given K, K0 ∈ K, K ≡ K0 provided, for all idempotents d
of Z(Λ), we have ddK = dK if and only if ddK0 = dK0.
Let δK be the unique block of Λ such that δKdK = dK. So we have δK =
P
K0
dK0
that K ≡ K0. Now using Lemma 4.17, (Kinf1.δK)(e11e K K) = (δK.Kinf1)(e11e K K) = δK(eKK) + X HKK δKeKHe K K = (δKdK)(eKK) + X HKK (δKdH)eKHe K K = dK(eKK) = e K K 6= 0.
Therefore 0 6= δK(e11eKK) = δK.d1(e11eKK). In particular, δK.d1 6= 0. Therefore
K ≡ 1 for arbitrary K ∈ K. The equivalence relation ≡ has a unique equivalence class.
4.4
The Case K = {1, C
2, V
4}
In this section we shall show that Λ has only one block and Λdef has two blocks
for K = {1, C2, V4}. Let V4 = {1, a, b, c}, A = {1, a}, B = {1, b} and C = {1, c}.
For K = {1, C2, V4}, we have the basis e11, e C2 1 , e C2 C2, e V4 1 , e V4 A, e V4 B, e V4 C , e V4 V4 for the
Burnside algebra. We will write inductions, restrictions, isogations, inflations and deflations below. All basis elements goes to zero unless stated otherwise.
C2ind1 = n e1 1 → 2e C2 1 , V4ind1 = n e1 1 → 4e V4 1 V4indA= eC2 1 → 2e V4 1 eC2 C2 → 2e V4 A , V4indB = eC2 1 → 2e V4 1 eC2 C2 → 2e V4 B , V4indC = eC2 1 → 2e V4 1 eC2 C2 → 2e V4 C 1resC2 = n eC2 1 → e11 , 1resV4 = n eV4 1 → e11 AresV4 = eV4 1 → e C2 1 eV4 A → e C2 C2 , BresV4 = eV4 1 → e C2 1 eV4 B → e C2 C2 , CresV4 = eV4 1 → e C2 1 eV4 C → e C2 C2
1iso1 = n e1 1 → e11, C2isoC2 = eC2 1 → e C2 1 eC2 C2 → e C2 C2 , V4isoV4 = eV4 1 → e V4 1 eV4 A → e V4 A eV4 B → e V4 B eV4 C → e V4 C eV4 V4 → e V4 V4 C2inf1 = n e1 1 → e C2 1 + e C2 C2 , V4inf1 = n e1 1 → e V4 1 + e V4 A + e V4 B + e V4 C + e V4 V4 V4infV4/A= eC2 1 → e V4 1 + e V4 A eC2 C2 → e V4 B + e V4 C + e V4 V4 , V4infV4/B = eC2 1 → e V4 1 + e V4 B eC2 C2 → e V4 A + e V4 C + e V4 V4 V4infV4/C= eC2 1 → e V4 1 + e V4 C eC2 C2 → e V4 A + e V4 B + e V4 V4
There is no short formula for deflations for the primitive idempotent basis. There-fore, we computed deflations by the transitive G-set basis and then passed to the primitive idempotent basis by using the transformation formula by Gluck [5]. For this, we need the table of marks and the inverse of it.
MG(I, U ) 1 A B C V4 1 4 2 2 2 1 A 0 2 0 0 1 B 0 0 2 0 1 C 0 0 0 2 1 V4 0 0 0 0 1
1defC2 = eC2 1 → 1/2e11 eC2 C2 → 1/2e 1 1 , 1defV4 = eV4 1 → 1/4e11 eV4 A → 1/4e11 eV4 B → 1/4e 1 1 ev4 C → 1/4e 1 1 V4/AdefV4 = eV4 1 → 1/2e C2 1 eV4 A → 1/2e C2 1 eV4 B → 1/2e C2 C2 eV4 C → 1/2e C2 C2 ,V4/BdefV4 = eV4 1 → 1/2e C2 1 eV4 A → 1/2e C2 C2 eV4 B → 1/2e C2 1 eV4 C → 1/2e C2 C2 V4/CdefV4 = eV4 1 → 1/2e C2 1 eV4 A → 1/2e C2 C2 eV4 B → 1/2e C2 C2 eV4 C → 1/2e C2 1
Also we can give them in matrix form. Let ai,j represent the entry in the i’th row
and j’th column of the corresponding matrix. All entries of the matrices below are zero unless stated otherwise.
C2ind1 =a(2,1) = 2 , V4ind1 =a(4,1) = 4 , V4indA=a(4,2) = a(5,3) = 2 , V4indB =a(4,2) = a(6,3)= 2 , V4indC =a(4,2) = a(7,3) = 2 , 1resC2 =a(1,2) = 1 , 1resV4 =a(1,4) = 1 , AresV4 =a(2,4) = a(3,5)= 1 BresV4 =a(2,4) = a(3,6) = 1 , CresV4 =a(2,4) = a(3,7)= 1 1iso1 =a(1,1) = 1 , C2isoC2 =a(2,2) = a(3,3)= 1 and V4isoV4 =a(4,4) = a(5,5) = a(6,6) = a(7,7) = a(8,8) = 1
C2inf1 =a(2,1) = a(3,1) = 1 V4inf1 =a(4,1) = a(5,1) = a(6,1)= a(7,1) = a(8,1) = 1 V4infV4/A =a(4,2) = a(5,2) = a(6,3)= a(7,3) = a(8,3) = 1 V4infV4/B =a(4,2) = a(5,3) = a(6,2)= a(7,3) = a(8,3) = 1 V4infV4/C=a(4,2) = a(5,3) = a(6,3)= a(7,2) = a(8,3) = 1 1defC2 =a(1,2) = a(1,3) = 1/2 1defV4 =a(1,4) = a(1,5) = a(1,6)= a(1,7) = 1/4 V4/AdefV4 =a(2,4) = a(2,5) = a(3,6)= a(3,7) = 1/2 V4/BdefV4 =a(2,4) = a(3,5) = a(2,6)= a(3,7) = 1/2 V4/CdefV4 =a(2,4) = a(3,5) = a(3,6)= a(2,7) = 1/2
By Corollary 4.10, we know that every idempotent of Z(Λ) belongs to Z(Υ). So every idempotent of Z(Λ) is a sum of dk’s for k = 1, 2, 3.
Here d1 = diag(1, 1, 0, 1, 0, 0, 0, 0) , d2 = diag(0, 0, 1, 0, 1, 1, 1, 0) , d3 =
diag(0, 0, 0, 0, 0, 0, 0, 1). We have to find which of those commute with inflations. Now take an idempotent δ from Λ. Then δ has the form δ = ad1+ bd2+ cd3where
a, b ∈ {0, 1}. Commutativity withV4inf1 gives us a = b = c. So only idempotents
are 0 and 1. It has just one block. However that is not the case for Λdef.
Theorem 4.19. Λdef has two blocks for K = {1, C 2, V4}.
Proof. By Corollary 4.10, we know that every idempotent of Z(Λdef) belongs to
Z(Υ). So every idempotent of Z(Λdef) is a sum of dk as above. We have to find
which of those commute with deflations.
Now take an idempotent δ which δd1 = d1. Then δ has the form δ = d1+ad2+bd3
where a, b ∈ {0, 1}. Commutativity with V4/AdefV4 gives us a = 1. So every
idempotent that contains d1 should also contain d2. However no other deflation
gives any further restriction. Therefore we have δ = d1+ d2 or δ = d1+ d2+ d3.
a, b ∈ {0, 1}. If we check commutativity with deflations for this δ we get no restriction except that if it contains d1, it has to contain d2 too. Therefore we
have another idempotent d3.
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